Real-rooted atlas

A059364 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n x^{2} + 2\,n^{2} x + 2\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
27411
3465818122
4326658438961518
5255672087563349660021423
6222128112811781384976304204320340869
721297695999217943481779396986900290088352806058980
8223934412047544276690363515427626676060120850323025008322560119218860
9256593601607836324393322046835711926622462054089517681571429163433996832659202575293165
1031854096022818071527244408724133770458321582934815512445423908649967603621736586884222497603628800060628447215
114261576320343895869441245827226962667758779363738865580503582924825422377123859181077941172803196602338455962864004390848001545696702270
12611485113605493424619522238587960040546140997630488621634236901004176870896681063557725224661379102944187077888727249896536832079579082880574801920042437227275450
13937030429440928065576268842052409448624115261930975616212818200046600278834511335088265789139567959185696593204400943635738023643400787686152082493119272961209048353280809512704001248581232985275
1415275952518400165446688464640825679912745808251390727434188852093567927722807757863430809200854113411199248970380751311533444340174950004760197799596762624064746098257684814412521655244819551502414080122049607680039197268410049225
Technical details
Node id
n-e97240c990f04bb2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-61c5fbc18aed5547

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        1;
  2:        7,         4;
  3:       46,        58,        18;
  4:      326,       658,       438,        96;
  5:     2556,      7208,      7563,      3496,       600;
  6:    22212,     81128,    117813,     84976,     30420,      4320;
  7:   212976,    959992,   1794348,   1779396,    986900,    290088,     35280;
  8:  2239344,  12047544,  27669036,  35154276,  26676060,  12085032,   3025008,   322560;
  9: 25659360, 160783632, 439332204, 683571192, 662246205, 408951768, 157142916, 34339968, 3265920;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(2*n^3-2*n^2)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.